You are docking with a space station that is rotating about the axis of the dock

Question

You are docking with a space station that is rotating about the axis of the docking port at a constant angular speed omega_s. There is also a red marker beacon attached to the space station. In order to successfully dock, the spacecraft tail must be aligned with the marker beacon. Your thrusters will provide a constant torque T about the space craft longitudinal axis. Your space craft is initially not spinning about its longitudinal axis, and it it has a mass moment of inertia I_GG about its longitudinal axis. Find the length of time t required to fire the thrusters in order to spin the space craft up to the space station angular speed omega_s. Find the angle theta at time = theta (the time you fire the thrusters) so that when you have spun the spacecraft angular speed up to the space station speed, the marker beacon and the tail are aligned.

Explanation / Answer

here,

Since , Translational Torque
t = T*r ( T is tension, r is radius)

Also, Rotational Torque
tr = I*alpha ( alpha = angular acceleration, I is moment of inertia)

Therefore,
Translational Torque = Rotational Torque
T*r = I*alpha
T*r = I*ws/t ( Since alpha = Ws/t)

Solving for time , t = I*Ws/T*r -----------------(1)

also, From Third Eqn of Rotational Motion,
ws^2 = 2*alpha*Theta

Theta = ws^2/(2*alpha) ----------------------------(2)

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